For many purposes, zeolites and related materials are not utilized in the as-synthesized form. Rather, they are only employed after an appropriate post-synthesis modification.Undoubtedly, the classic procedure of zeolite treatment after synthesis is thatof ion exchange achieved through treatment of a suspension of the as-synthesized (or natural) zeolite powder (usually in the sodium or potassium form) inan aqueous solution of a salt containing the cations to be introduced. Starting inthe 1930s, this type of ion exchange has been extensively studied, not only as amethod of preparation, but also with respect to thermodynamics and kinetics.Application on an industrial scale is well developed and, because of its importance, ion exchange in zeolites has been reviewed several times. Thus, thefirst chapter of Volume 3 of the series “Molecular Sieves – Science and Technology”, which was contributed by R.P. Townsend and R. Harjula, was able to focuson the developments and advances made during the last decade. It emphasizesthe need for improvement of theoretical approaches, utilization of the rapidlygrowing computational power, and the importance of acquiring reliable data asthe bases for progress in fundamental studies on conventional ion exchange.The more recent development of solid-state ion exchange and related modification techniques such as reactive ion exchange between solid zeolite powdersand solid or gaseous compounds containing the cations we wish to introduce israther exhaustively dealt with in the subsequent chapter written by H.G. Kargeand H.K. Beyer. The concept of solid-state ion exchange is explained and contrasted to the conventional exchange process. Experimental procedures as wellas techniques for monitoring the solid-state modification of zeolites are described in great detail and illustrated by a large number of investigated systems.Related methods of post-synthesis modification, possible mechanisms, and firstapproaches to study the kinetics of solid-state ion exchange are discussed.

Post-synthesis modification of zeolites via alteration of the aluminum content of the framework became a most important topic of zeolite chemistry when,in the mid 1960s, the effect of stabilization through dealumination was discovered. In Chapter 3, H.K. Beyer contributes a systematic review on techniquesfor the dealumination of zeolites by hydrothermal treatment or isomorphoussubstitution amended by a section on the reverse process, i.e., introduction ofaluminum into and removal of silicon from the framework.Methods of post-synthesis modification essentially different from those discussed in the first three chapters are based on the generation of extra-frame-

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Preface to Volume 3

work aggregates of metals (as presented in the chapter by P. Gallezot), ionic clusters (as described in the contribution by P.A. Anderson), and oxides and sulfides (treated in the last chapter written by J. Weitkamp et al.). One of the mainmotivations for studying the generation of such clusters inside the void volumeof zeolite structures originates, of course, from possible applications in catalysis.This is most evident in the case of metal cluster/zeolite systems which are successfully employed in heterogeneous catalysis of hydrogenation, hydrocracking,hydroisomerization, etc. However, both ionic clusters and oxidic and sulfidicclusters hosted by the frameworks of zeolites are interestring candidates as catalysts for base-catalyzed, redox, photocatalyzed and perhaps other reactions. Inview of cluster formation with zeolites as hosts, questions of size, location, distribution, interaction with the framework, and stabilization of the active aggregates play a decisive role. Thus, in all three contributions on clusters in zeolites,methods of their preparation as well as problems of their characterization andutilization as catalysts and photosensitive materials, as sensors, in optics, andelectronics are extensively dealt with. These areas are still challenging for futureresarch and promising in view of potential applications.However, not all important phenomena of post-synthesis modification arecovered with the present six chapters of Volume 3 of the series ‘Molecular Sieves– Science and Technology’. Topics such as, for instance, ‘Incorporation of Dyesinto Molecular Sieves’, ‘Preparation of Ship-in-the-Bottle Systems’, ‘SecondarySynthesis in Zeolites’, ‘Pore Size Engineering’, ‘Modification of MesoporousMaterials’ are equally important and, to a large extent, presently subject to veryactive research and development. Therefore, such topics will be dealt with in oneof the subsequent volumes under the title ‘Post-Synthesis Modification II’.September 2001

Throughout the 1990s there was a decline in the number of fundamental studiescarried out on the ion exchange properties of zeolites and related materials. Onehas only to examine the content of published conference proceedings on thesubject over the last 20 years to observe this trend: the situation has moved fromone where whole sessions were devoted to ion exchange studies, to one wherethe subject is subsumed into sessions covering other areas. Part of this decline isto be expected, as increased attention has been rightly paid to the intriguingpossibilities that can arise through the exploitation of newer alternative postsynthesis methodologies, many of which are discussed elsewhere in this volume.Nevertheless, the fact remains that conventional ion exchange techniquescontinue to be used routinely for post-synthesis modification during the preparation of molecular sieves for major industrial applications. Also, there are nowareas where molecular sieves find major application directly as ion exchangersper se. In this respect the situation has changed markedly since the early 1960s,when Helfferich, in his classic book on ion exchange, could justifiably describezeolites “as ion exchangers they are of little practical importance” [1]. Thesedirect applications are especially detergency [2–7] and also the removal ofnuclear waste [8–13] or other environmental pollutants [3]. However, it isgenerally a combination of properties of a particular zeolite in addition to itsion exchange capability that has tipped the balance in favour of its use, ratherthan any intrinsic superiority per se, which the zeolite may possess as an ionexchanger.If, therefore, conventional ion exchange remains an important post-synthesispreparative technique, and the materials have in addition major direct applications as ion exchangers, why have the number of fundamental studies decreased?It is certainly not because ion exchange behaviour of molecular sieves is sufficiently well understood and predictable to render further fundamental researchstudies unnecessary. Two causes are suggested to explain this decline:1. Many theoretical treatments of the ion exchange reaction within zeolites(both equilibrium and kinetic) are obscure and complicated. This has without doubt rendered inaccessible the real value of the work to those manyworkers who have a practical need to predict and control ion exchange behaviour during the industrial exploitation of molecular sieves. Althoughtheoretical understanding is important, it is easy to forget that the end purpose of such work should be to provide information and tools that thechemical engineer or other user of the molecular sieve can apply simply andeffectively. Obscurities in theoretical treatments mean that users often do notappreciate how basic theory can be used, not just to simplify the number ofmeasurements which need to be made, but also to predict and control behaviour during application. The theory should not be an end in itself!

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2. The second cause is related to the first. Even where the value of theory for theprediction and control of the behaviour of these materials has been recognised, the utility of these approaches has often been greatly reduced becauseof the experimental methods which have been employed or by the poorexperimental data which have been available, or both. Indeed, it is only comparatively recently that a proper recognition has arisen concerning the number of potential pitfalls and difficulties that can militate against the acquisition of meaningful and accurate experimental data.A good example of this is the frequently studied Na/Ca-zeolite A system, whichhas received much attention because of its importance in detergency applications. Careful and detailed experimental studies over a period spanning some20 years by different sets of workers [14–20] resulted in calculated values of thestandard free energy of exchange (kJ equiv–1*) which ranged from –0.59 [14] to–3.09 [17]. Plots of the corrected selectivity coefficient (defined below; see

* Throughout this paper the term “equiv” denotes 1 mol of unit negative or positive charges.

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Eq. 7b) naturally show a similar diversity but also differ from each other in curveshape and trends (Fig. 1). These marked differences (particularly at the extremaof the plots) were variously ascribed to experimental error [20], to variablequantities of non-exchangeable sodium in the materials employed [20] (thematerials differed in their source and in their method of preparation [14–20])or to variable levels of hydronium exchange depending on the pH and other conditions used [20, 21].Thus, even for this very important example, not only is some of the publishedtheoretical work difficult to interpret, but also experimental data from differentstudies are frequently incompatible and incomplete.It is essential therefore that a critical review of advances over the last decadeshould look at the developments in the context of the field as a whole. This is ourintention here. After a discussion of the origin, ubiquity and nature of ionexchange behaviour in molecular sieves, recent advances in the application ofthermodynamic and kinetic descriptions of the ion exchange process willbe described. This will demonstrate some of the shortcomings of currentapproaches, together with the relative paucity of reliable literature data thatcan be applied easily and practically. This whole topic has particular relevance to those industrial applications where zeolites are used directly as ionexchange materials and this will be exemplified throughout the chapter usingtwo main examples. The first of these is the application of A- and P-type zeolitesas detergent builders, where the approach is to use a batch exchange approachto remove hardness ions (especially calcium) as fast as is practicable beforethe indigenous water hardness harms the wash performance of the detergentproduct. The second concerns the treatment of nuclear waste, where a varietyof higher silica zeolites have been employed using a continuous (column)process to remove, and subsequently store, high concentrations of monovalentand divalent radionuclides such as caesium and strontium. For both thesemajor applications, in addition to selectivity, it is noteworthy that the systemsare normally multicomponent, that the kinetics of exchange are all important and that the morphology of the exchanger material must be controlledcarefully.Post-synthesis modification comes into its own when preparing molecularsieves with desirable and exploitable properties other than those of ion exchange, be they optical, magnetic, catalytic or adsorptive. Here it is not directlythe thermodynamic and kinetic ion exchange properties that are of primeimportance but rather which experimental, preparative methods are most commonly used. Thus it is important to assess what are the most appropriate experimental methods of preparation, as well as to review the many pitfalls one canfall into which can subsequently give rise to very inaccurate and inadequateexperimental data. These experimental problems can include frameworkhydrolysis, hydronium exchange, dealumination, the presence of key trace impurities, dissolution phenomena, carbonate and bicarbonate interference, colloidalphenomena, metal ion complex formation and cation hydrolysis.Having thus reviewed developments and advances over the last decade, thechapter concludes with some recommendations on directions and topics for thisarea of research in the future.

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1.2Origin and Nature of Ion Exchange Behaviour in Molecular Sieves

Ion exchange is a characteristic property manifested by most molecular sieves.In essence, whenever isomorphous replacement of one cation by another of different charge occurs within an initially neutral crystalline framework such as apure silica molecular sieve, then a net electrical charge remains dispersed overthat framework. This is neutralised through the presence, within the microporous channels, of cations of opposite charge (often referred to as counterions).An example of this is seen in the introduction by direct synthesis of small quantities of aluminium into the silicalite framework to give the material ZSM-5.Silicalite, the pure silica analogue of ZSM-5, is then seen to be just the end-member of a set of isomorphous microporous molecular sieves that exhibit ionexchange properties which are a function of the quantity and distribution ofaluminium atoms within the structurally similar frameworks. In addition, sinceone can prepare, through post-synthesis modification of the framework composition, a variety of other isomorphous metallosilicates and metal aluminosilicates, it is obvious that zeolites possessing ion exchange capabilities are acommon occurrence.Pure aluminium phosphate molecular sieves are probably more commonthan are pure silica analogues of zeolites. They resemble pure silica zeolites inthat they possess frameworks that are electrically neutral, but there is a significant difference between these two classes of inorganic solids. In topologicalterms both are 4:2 connected nets of T:O atoms (“T” denoting tetrahedralframework and “O” denoting oxygen). From this it is obvious that it is onlyrequired for the T ion to have a charge of +4 for the connectivity of the net togive rise naturally to a neutral framework in concert with the oxide anions. Thisis fulfilled for pure silicalite. In the case of ALPO molecular sieves the requirement is also fulfilled, but the 4:2 T:O net now comprises two types of strictlyalternating T-cations (aluminium and phosphorus, possessing respectively formal positive charges of 3 and 5). Providing the cations alternate strictly throughout the framework, the 4:2 Al,P:O net holds no overall charge; however, incontrast to a pure silica zeolite, where the formal charge at every atomic centreis zero, within a pure AlPO the formal charge is not dispersed homogeneously,but changes from –1 at each aluminium to +1 at each phosphorus. This greaterheterogeneity of charge distribution may in part explain the experimentalobservation that ALPOs frequently exhibit poorer thermal stability than do puresilica zeolites.For a particular ALPO molecular sieve to possess an ion exchange capacity asan intrinsic property, it is necessary to prepare a material where some of the aluminium and/or phosphorus framework atoms have been replaced by otheratoms of different charge. This can occur using for example silicon, to form theso-called SAPO materials, or with metals in addition or not to silicon, to formrespectively the so-called MeAPSO and MeAPO analogues. However, it is important to note that although silicon could in principle replace either aluminium orphosphorus to give rise to positively or negatively charged SAPO molecularsieves, respectively, in practice only the latter process seems to occur, or another

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process in which two silicons replace one of each of aluminium and phosphorus,which gives rise to no net change in framework charge [22]. In MeAPSOs, divalent or trivalent metal ions replace the aluminiums in the framework. In this waythe charge imbalance is minimised as these isomorphous substitutions eithermake no difference to the overall framework charge (T3+ for Al3+) or onlyincrease it by one negative charge per substitution (e.g. Mg2+ for Al3+), a processanalogous to when aluminium replaces silicon in aluminosilicates [22].Overall therefore, and in common with aluminosilicate zeolites, the norm isfor MeAPSOs and MeAPOs to possess cation exchange properties rather thanthe reverse. In this respect, zeolites and ALPOs resemble many other classes ofion exchangers that are mineralogical in origin, such as the clay minerals. Theseare layered materials where a cation exchange property can arise primarily fromisomorphous replacement of trivalent cations by divalent, or tetravalent cationsby trivalent ones, within the layers [23]. However, there is a major exception:these anionic exchangers are the double metal hydroxides, which are also layered structures but which exhibit a net positive charge across the lattice. The“parent” material here is the mixed Mg,Al hydroxide, commonly referred to ashydrotalcite. It would be intriguing to understand better the conditions (if any)under which one might expect to synthesise microporous three-dimensionalframework structures which similarly have a net positive charge dispersed overthe lattice and hence an anion exchange capacity coupled with a molecular sievecapability.It is important to note that, up to this point, we have been considering the zeolite, ALPO, SAPO, etc., as being described adequately as a 4:2 T:O net. This topological description, which in general terms is, as Smith points out [24], nothingmore than a mathematical construct of the human brain, does neverthelessallow us to appreciate both the origin and magnitude of an ion exchange capacity arising from T-atoms being replaced by others of different charge. However,this description is not sufficient to cover the observed differences in ionexchange properties (i.e. selectivity, kinetic rate, level of exchange) that may beseen between various molecular sieves having similar exchange capacities. Tounderstand these differences, one must not only examine more closely the topological properties of the nets but also bring to bear structural considerations.Considering these topological properties in more detail, it is adequate at thispoint to take as read that all the T-atoms within the microporous net are joinedto each other by bridging oxygens. One can therefore concentrate on the Tatoms only and describe molecular sieves in terms of four-connected threedimensional (4-conn.3D) nets of T-atoms [25] that, in turn, can be derived fromappropriate 3-conn.2D nets [26]. Considering the latter nets first, these differfrom one another in the ways the nodes (T-atoms) link to each other via networks of polygons. Any node can then be described by its “vertex symbol”, viz.by its surrounding polygons with the number of each type of similar polygonsurrounding the node being denoted by a superscript [26]. Thus the simplestexample of a 3-conn.2D network (the hexagonal net) becomes a 63-net; a morecomplicated example could be the 4.6.12-net which forms the basis for thegmelinite structure [26]. Note that all the nodes within each of these two separate examples are topologically equivalent. This need not be the case. For exam-

Ion Exchange in Molecular Sieves by Conventional Techniques

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Fig. 2. Structure of mordenite viewed along the main 8-ring and 12-ring channels parallel tothe c-axis. Four topologically distinct types of T-atoms are observed within the 3-conn.2D(4.5.8)1 (4.5.12)1 (5212)1 (5.8.12)1 net

ple, consider the case of mordenite (Fig. 2), which is derived from a (4.5.8)1(4.5.12)1 (5212)1 (5.8.12)1-net containing four topologically distinct types of Tatoms [24].Similar considerations apply when one considers the 4-conn.3D nets thatconstitute molecular sieves. Here it is often convenient to describe the structurein terms of polyhedral units or cages, with the polyhedra described topologically in terms of face symbols [25] (not to be confused with vertex symbols definedabove). Thus the face symbol for the familiar sodalite unit, which is geometrically a truncated octahedron, is 4668 with all vertices geometrically and topologically equivalent. If these units are then linked together, for example eitherthrough their 4-windows or half their 6-windows, one forms respectively thezeolite A and faujasitic structures. Both these structures possess cubic symmetry, with each structure comprising 26-hedral cages connected to each otherthroughout the microporous zeolite framework, but the vertices of the sodaliteunits are no longer all topologically equivalent. For zeolite A the sodalite unitsenclose a cage which is the great rhombicuboctahedron (4126886) [25] whereasfor faujasite the cage is the so-called 26-hedron type II, denoted by the face symbol 4641264124 [25].Why are these matters significant when one considers the ion exchange properties of molecular sieves? The answer is that these topologically non-equivalentT-atoms combined with the overall structural properties of the three-dimensional microporous framework often give rise to several very different types oflocal environments which repeat themselves regularly throughout the crystalline structure. These different local environments, evidenced by solid stateNMR combined with X-ray crystallography [27], are distinct in themselves, differing from each other sterically and electronically, and these differences will be

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manifested not only through their characteristic adsorptive and catalytic behaviour, but also through their ion exchange properties. Formally, therefore, zeolitesmay be regarded as comprising a set of crystallographically distinct sublattices,each having characteristic selectivities for different exchanging cations, depending on these local environments [28]. The overall ion exchange behaviour of amolecular sieve can therefore be a subtle function of the structural and topological properties combined. An important combination of structural and topological properties concerns the ordering of isomorphously substituted framework atoms [29]: this determines what fraction of the overall framework chargeis found on each sublattice. Other significant structural properties can be lossesin symmetry through restricted rotation [27], and whether the sites are accessible to exchanging cations (i.e. the sizes of the micropore channels allowingingress and egress of exchanging cations plus water).A further point is worth emphasising: since site heterogeneity in a particularzeolite is manifested through such a set of crystallographically distinct sublattices, zeolites differ in this respect significantly from some other common classes of ion exchangers, such as the clay minerals or the resins. Whereas in zeoliteswell-defined sites are repeated regularly through the crystalline matrix, in clayminerals and resins site heterogeneity is often manifested in terms of patches, orregions of the surface where the sorption energies are approximately constant[30]. Thus a statistical thermodynamic model of ion exchange for clay mineralsand resins [30] can differ markedly in character from ones developed for zeolites[31, 32].As a consequence of all these factors combined, both the equilibrium andkinetic aspects of selectivity and uptake of ions within molecular sieves canrarely be understood in a straightforward manner. Phenomena which havereceived either considerable attention in recent years or deserve further studyinclude the so-called “ion sieve effect”, behaviour of high silica materials, theeffects that framework flexibility can have on selectivity and rates of exchange,multicomponent ion exchange, prediction of exchange equilibria, and the possibility of inducing phase transitions within zeolites through ion exchange. Manyof these are considered further below.So far we have considered topological and internal structural factors whichgive the molecular sieve particular ion exchange properties. However, an ionexchange capacity can also be manifested which is not an intrinsic property ofthe material. The source of this property is unsatisfied valencies occurring at thetermination of the crystal edges and faces, or at faults within the crystallinestructure. In formal terms, the origin of this is topological, in that this incidental and secondary property arises from disruptions in the net at interfaces, surfaces and faults, but the nature and extent of this incidental property dependsessentially on structural and morphological characteristics. For the former, wecan take as an example an ion exchange capacity arising either from the presence of silanol groups [33, 34], or from hydroxyl groups attached to aluminiumatoms situated at the surface [35]. In clay minerals, as much as a fifth of the totalexchange capacity may arise from such sources whereas in the case of zeolitesthe contribution of such incidental (or secondary) ion exchange properties isusually small compared to the intrinsic, or primary source. The exception here

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can be high silica zeolites [35, 36], whose overall ion exchange properties havereceived considerable attention over the last decade [37–42].Interestingly, the external morphology can also be an important factor indetermining ion exchange behaviour of molecular sieves. The crystal habit, theaverage crystallite size, the distribution of crystallite sizes and the properties ofaggregates of crystallites can all affect the magnitude of secondary ion exchangecharacteristics, since these can alter significantly the surface to volume aspectratio and hence the number of external surface sites available [35]. Also, thekinetic properties may depend on these morphological characteristics, asinstanced by recent studies on a highly aluminous form of zeolite P [6, 7].

2The Importance and Utility of Theoretical ApproachesWhen a zeolite in (say) the sodium-exchanged form is suspended in a solutioncomprising a mixture of different cations and anions, two properties of thematerial are brought into sharp focus. The first of these concerns which types ofcations are “preferred” over sodium or each other by the zeolite. This property iscommonly referred to as the selectivity of a given form of zeolite for anothercation, but there are so many definitions of “selectivity” that the term “preference” may be better used for the present. The second key property to which one’sattention is drawn, and which is separate from selectivity (however defined), isthe rate at which the mixture of cations achieves its equilibrium distributionbetween the exchanging phases (viz., the electrolyte solution and the sublatticeswithin the zeolite).2.1Preference, Uptake and Selectivity

The preference manifested by a molecular sieve for a particular cation is strongly dependent not only on the character of the material under examination, butalso on the conditions of the system as a whole (viz., temperature, perhaps pressure, composition of exchanger and solution phases, pH, nature of solvent, etc.).Given a comprehensive definition of these conditions, the preference of a givenform of zeolite for a given cation will then be invariant for that set of conditionsbecause it is essentially an equilibrium property of the system. However, it isimportant to define clearly what is meant by “preference”. There are numerousselectivity coefficients defined in the literature and, on occasion, “selectivitycoefficient” is confused with “separation factor”, a function whose value doesdepend strongly on the total ion concentration in solution. Similarly,“uptake” or“loading” is often confused with “capacity”. To distinguish these terms, a fewbasic definitions are required.Considering as an example a binary exchange involving cations A (valencyzA) and B (valency zB), the reaction equation is usually written as:––(1)zA B zB + zB AzA = zA BzB + zB AzA

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where the overbars denote the exchanger phase. The preference displayed by thezeolite for one ion over another is then described by a selectivity coefficient,which is just a mass action quotient. According to the choice of concentrationunits, a series of these selectivity coefficients may be defined which differnumerically from one another:––x zB c zAEAzB cBzAc–AzB cBzAA BxEkA/B = 0≠ kA/B = 01(2)––x zAc zB ≠ k A/B = 01cAzB c–BzAEBzA cAzBB Awhere cA , cB are the cation concentrations in solution (mol dm–3) and the corresponding concentrations in the molecular sieve are indicated with an overbar(equiv kg–1 dry exchanger). The definition of kA/B is consistent with IUPAC recommendations [43] but is not very convenient for zeolites because of the signifX and k E are selectivity coefficients in which the zeoliteicant water content. kA/BA/Bphase cation concentrations are defined in terms of the mole fraction and equivalent fraction E, respectively:––(3a)X A = c–A / Si c–i .–(3b)E = z c– / S z c– .A

A A

i i i

When zA = zB = zi , then equivalent and mole fractions are numerically identicalE . Otherwise, these functions are not numerically identical. In pracand k XA/B = kA/BEtice, kA/B has been used most extensively for studies on zeolites.The selectivity coefficients given in Eq. (2) can be used to derive more fundamental equilibrium properties of the system, such as the standard thermodynamic functions describing the exchange reaction (viz. DGq, DHq, DSq), providedone has information on the nature and extent of all activity corrections for nonideality. However, the key point to note is that having defined the referencestates, by contrast with a selectivity coefficient, these standard thermodynamicfunctions are independent of exchanger composition since they refer by definition to a reaction between components which move from one set of specific,defined standard states to another. The magnitudes and signs of these standardfunctions therefore give no immediate information whatsoever on the actualpreference which a zeolite may display for a particular ion under a given set ofexperimental conditions. This point, obvious to the thermodynamicist, hasoften been missed, and effort has been invested uselessly in attempting to relatecalculated values of standard thermodynamic functions to mechanistic theoriesof exchange under real conditions. This has resulted in work being publishedthat is of little practical utility, if not plainly wrong. The issue of misunderstanding and consequently misusing thermodynamic data in this manner isexpanded elegantly by McGlashan [44].The selectivity of a particular molecular sieve for a given ion as a function ofexchanger composition is normally measured from an ion exchange isotherm,which is an isonormal [45], isothermal and reversible plot of equilibrium distributions of ions between the solution and zeolite phases. It is emphasised that itis only valid to calculate selectivity coefficients, and derived thermodynamicdata, from isotherms which are reversible (that is, the forward and reverseisotherms coincide within experimental uncertainty). The types of isotherms,

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Fig. 3. Examples of ion exchange isotherms exhibiting both unselective and selective behaviour towards the incoming ion A (curves are respectively convex and concave with respect tothe ordinate). Clear limits to exchange are also observed which are lower than those expectedon the basis of the theoretical exchange capacity of the zeolite. The arrows depict reversiblebehaviour

Fig. 4. Example of an ion exchange isotherm showing non-reversibility of exchange within aplateau region, characteristic of phase separation and coexistence of two phases over the composition range corresponding to hysteretic behaviour

and the causes for the shapes observed, are discussed elsewhere [45]. However,two isotherm types, which are particularly characteristic of molecular sieves(although not uniquely so), are shown in Figs. 3 and 4.Figure 3 shows isotherms for which only partial exchange for the incomingcation occurs. The isotherm plots enable one to distinguish clearly variousbasic definitions. Taking, for example, a constant level of exchange or uptake for–an incoming ion (e.g. EA = 0.5, then for this given uptake, the selectivity coefficient can vary from low to high values (cf. the two depicted curves). The abscis–sa of the isotherm ranges from EA = 0 to EA = 1; values of EA are determinedby dividing the uptake by the ion exchange capacity, which is the number ofexchange sites of unit charge per unit quantity of exchanger (defined as con-

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venient – see comments on this above). However, Fig. 3 shows curves which are–asymptotic to values of EA < 1, demonstrating that the maximum uptake (orloading) under specified experimental conditions for the incoming cation canbe less than what would be expected from the value of the ion exchange capacity. The cause of this may be due to inadequate experimental rigour, especiallyduring batch exchange experiments (see Sects. 2.2 and 3.2.1 for further discussion); however, genuine “ion sieve” or “volume steric” effects can also operate asa consequence of the crystalline and microporous nature of molecular sieves.Ion sieving, known for a long time and commonly observed in zeolites, ariseswhen part of the microporous channel network within the molecular sieve isinaccessible to the incoming exchanging cations simply because their ionicdiameters exceed the free diameters of the windows through which they mustpass [46]. The “volume steric” effect is less common, and arises when thecations have free access to the microporous voids and channels within thecrystal but nevertheless the size of the incoming ions is such that the channelsare completely filled before 100% exchange for the incoming ion can beachieved [47].Over the last decade, during a series of studies on high silica zeolites including ZSM-5, ZSM-11 and EU1-1, another possible cause for partial exchange hasbeen identified. Although full exchange of hydronium ion for sodium wasobserved by Chu and Dwyer for a range of high silica zeolites [37], and ion sieveeffects were identified by the same workers to explain partial exchange withsome organic-substituted ammonium cations in ZSM-5 [39], Matthews and Reesfound more complex behaviour with alkaline earth and rare earth cations inZSM-5 [38]. Univalent cations exchanged to 100% but this was not the case formultivalent cations. Part of the explanation for the significantly lower maximumloadings found with multivalent cations (especially Ca2+ and La3+) was ascribedto the distribution of the relatively low number of aluminium atoms in theframework, which could make it difficult for multivalent cations to neutraliseeffectively widely spaced negative charges on the framework [38]. To test thishypothesis, McAleer, Rees and Nowak [40] carried out a series of Monte-Carlosimulations which implied that the charge on divalent cations could only be satisfied adequately by aluminium atoms within the framework which were spacedapart by < 0.12 nm. More recently, similar experimental and theoretical studieswere carried out on zeolite EU-1, where analogous behaviour to ZSM-5 wasobserved, although cut-off values for exchange were much higher in EU-1 [41].Topological and structural differences between ZSM-5 and EU-1 were proposedas explanations for this different behaviour [41] (see the earlier discussion inSect. 1.2).Figure 4 shows a type of isotherm shape that is seen with crystalline ionexchangers such as molecular sieves and clay minerals, but is nevertheless relatively uncommon. The shape resembles the type II vapour adsorption isothermof the Brunauer classification, having a clear “plateau” region and inflexionpoint. An example is the Na/K exchange in zeolite P [48] that was found to bereversible over the whole range of equivalent fraction of potassium in the crys–tal (EK ). Zeolite P has the gismondine-type structure (GIS [49]). More commonly, isotherms of this type are found to be partially irreversible in the plateau

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13

region, resulting in a hysteresis loop between the forward and reverse isotherms(Fig. 4). Examples of such hysteretic behaviour include the Na/K and Na/Liexchanges in zeolite K-F [50], which is a framework structure isotype of edingtonite EDI [49], and the Sr/Na exchange in zeolite X [51]. Isotherms of this type(whether fully reversible or not) are characteristic of systems where the processof exchanging one cation for another induces structural distortions and changesin the molecular sieve framework, resulting in the end-members of the exchange––(EA = 0 and EA = 1, respectively) being different phases. If the framework is flexible and consequently the required structural transformation can occur readily,the plateau region (where the two phases coexist) will be reversible. This is thesituation observed for the Na/K exchange in zeolite P [48] which has long beenrecognised as a material which exists as several structural varieties [52] depending on ion exchange form and level of hydration [53] and which is recognised ashaving an unusually flexible framework [49, 52].When a hysteresis loop occurs, this corresponds to a situation where the endmembers of the exchange exhibit limited mutual solid solubility; in other words,over this region of the isotherm two separate phases coexist. Barrer and Klinowski considered the conditions under which phase separation may be expected to occur in a statistical thermodynamic treatment involving an interactionenergy for entering ions wAA/kT [31]. When this term is sufficiently negative, sothat the cations segregate rather than form a homogeneous phase, they showedthat conditions could arise under which a physical mixture of two A- and B-typecrystals has a lower free energy than the homogeneous A/B phase [31]. If inaddition the nuclei of the A-rich phase grow within the B-rich “parent” phasematrix then two positive free energy terms are involved in the exchange process.These are a strain free energy resulting from the misfit between the new growing phase within the old, and an interfacial free energy. These tend “to delay theappearance of the new phase beyond the true equilibrium points for forwardand reverse reactions” [31]. This is the proposed explanation for the hystereticbehaviour seen in systems such as the Na/K and Na/Li exchanges in K-F [50] orthe Sr/Na exchange in X [51], and contrasts with P [48, 53]. This has significancefor the use of a high aluminium analogue of P in detergency [6, 7]. This material, named “maximum aluminium P” (MAP), has the gismondine frameworkstructure of zeolite P but with a Si/Al ratio of unity [6]. The unusually flexibleframework [49, 52] is reported to lead to cooperative calcium binding, as well asto unusual water adsorption/desorption properties that enhance bleach stability[6, 7]. These properties, combined with superior kinetic behaviour, result in amaterial that reduces water hardness much more effectively than zeolite A (sic,[6, 7, 45]).2.2Batch and Column Exchange Operations

It is instructive at this point to compare these two techniques by consideringthe conversion of a zeolite from one ionic form (B) to another (A) as shown inEq. (1) and using the selectivity coefficient kA/B defined in Eq. (2).In batch ion exchange, a given amount m of zeolite in the B-form is contacted with a given volume v of a salt solution of ion A. At equilibrium, the ions aredistributed between the solid and solution phase according to:cAzBc–AzB=k.(4)5A/B 5cBzAc–BzAThe progress of the reaction is illustrated in Fig. 5 for two univalent cations(zA = zB = 1) assuming a constant selectivity coefficient kA/B = 10 and an ionexchange capacity of 4 mequiv g–1. It is clear that it is difficult to obtain a highdegree of conversion by a single batch equilibration. In this example, 430 cm3 of0.1 equiv dm–3 solution of ion A is required for 99% conversion. This is almostan 11-fold excess even though the exchange equilibrium operates in favour ofions A.In zeolites strong selectivity reversals are often observed and this makes itvery difficult to obtain a high conversion to the required ionic form. This problem is discussed in more detail in Sect. 3.2.1. Here, conversion will be discussedin qualitative terms. The solution concentrations of A and B [Eq. (4)] can bewritten as:–(5)cB = c–A /(V/m) = EAQ/(V/m)and–cA = cA(o) – cB = cA(o) – EAQ/(V/m)

(6)

Loading (meq/g)

Solution concentration (N)

where Q is the ion exchange capacity (equiv kg–1), V/m is the solution volume(dm3) to zeolite mass (kg) ratio in the batch equilibration and cA(o) is the initial

Solution volume (ml)

Fig. 5. Batch exchange: loading of ion A in zeolite (solid curve) and concentration of A in solution (broken curve) as a function of solution volume when contacting 1 g of zeolite in B-formbatchwise with 0.1 g equiv–1 solution of A. Selectivity coefficient kA/B and exchange capacity Qhave been given values of 10 and 4.0 mequiv g–1, respectively

15Outlet concentration (N)

Average loading (meq/g)

Ion Exchange in Molecular Sieves by Conventional Techniques

Effluent volume (ml)

Fig. 6. Column exchange: average loading of ion A in zeolite (solid curve) and concentrationof A in outlet solution (broken curve) as a function of solution volume passed through thecolumn. Mass of zeolite bed 1 g, inlet solution pure A at 0.1 equiv dm–3 concentration. kA/B andQ as in Fig. 5

concentration of A (equiv dm–3) in the solution. To obtain a high conversion tothe A-form in a single equilibration, kA/B and cA must be high and cB must be low.cB can be made low by using a large volume of solution per unit mass of zeolite(maximum value of cB = Q/(V/m)) (Eq. 5). cA can be made large by using a highinitial concentration of A and large V/m ratios (Eq. 6).In column exchange, a solution of ion A is passed through a column that contains a given quantity (m) of zeolite. This process is illustrated in Fig. 6 using thesame parameters as in Fig. 5 for the batch exchange. In column exchange, theconversion to the A-form proceeds much more easily, as ion B is constantlyremoved from the system. However, ion A is not homogeneously distributed inthe bed, but is first taken up by material near the column inlet and the conversion proceeds in the direction of solution flow. When most of the zeolite hasbeen converted to the A-form, ion A starts to emerge from the column and cAtends to the value of the feed concentration, when the column has become completely exhausted.The important point to note is that by contrast with batch exchange, far lesssolution is needed for full conversion. In the example of Fig. 6, only 50 cm3 of0.1 equiv dm–3 solution is required for every gram of zeolite to achieve 99% conversion. This is only a 25% excess.Figures 5 and 6 represent highly idealised cases and serve here only todescribe qualitatively the differences between batch and column exchanges. Inrealistic situations, the selectivity coefficient decreases with increasing loadingof A in the zeolite (see Fig. 1). This means that an even higher excess of A mustbe used under real conditions. In addition, in column exchange, the rate ofexchange reaction often tends to decrease at high loadings, which lowers thegradients of the loading and concentration curves (Fig. 5) and increases thesolution volume needed for full conversion.Pure synthetic zeolites are fine powders that are usually unsuitable for column operation. Therefore, batch methods are used for the study of ion exchange

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R.P. Townsend · R. Harjula

equilibria. Granular zeolite exchangers that are suitable for column work aremanufactured by using suitable binders (e.g. clay, silica, alumina) and care mustbe taken in extrapolating data obtained from batch experiments to columnoperation.2.3Thermodynamic Parameters, Non-Ideality and the Predictionof Exchange Compositions

To derive thermodynamic parameters of ion exchange, the normal procedure isto correct for solution phase non-ideality first by deriving a corrected selectivitycoefficient in which concentrations within the external solution are replaced byactivities. The means by which this may be done, for binary or multicomponentsystems, is described elsewhere [54, 55]. The corrected selectivity coefficientsE are then:corresponding to k XA/B and kA/B––XAzB aBzAKXA/B = 92,(7a)––XBzA aAzB–E zB a zAE = A B .KA/B(7b)–92EBzA aAzBE is identical to the function K shown in Fig. 1 and taken from [20].K A/BGThe thermodynamic equilibrium constant Ka is then obtained by integratingthe appropriate form of the Gibbs-Duhem equation to give as correspondingexpressions for Eqs. (7a) and (7b), respectively, the following:1

–X dElnKa – D = Ú lnKA/BA,0

(8a)

1

–E dElnKa – D = (zB – zA) + Ú lnKA/BA,

(8b)

0

where D is the water activity term [56, 57]. D is normally ignored on the assumption its magnitude is small; however, it should be noted that for the most commonly employed formulation, corresponding to Eq. (8b) and after Gaines andThomas [58], D π 0 when the system is behaving ideally if zA π zB but ratherequates to (zA – zB) [56, 59]. This must follow since, when the system is behavingideally, the values of all the activity coefficients are by definition unity for allE = constant [56, 57] sincecompositions and hence Ka = K XA/B = KA/B–g–AzBf AzBxEKa = KA/B=K(9)–5A/B 5g–BzAf BzAwhere fi , gi are the appropriate rational activity coefficients for cations in theexchanger phase in association with their equivalents of anionic charge.Equations (8a) and (8b) provide the starting point for the prediction of ionexchange equilibria in molecular sieves, an activity which has received a significant level of attention over the last decade or so. The basis for prediction comes

Ion Exchange in Molecular Sieves by Conventional Techniques

17

from a principle put forward some time ago [60], viz., that because D is small andchanges little with zeolite composition, and providing salt imbibition is negligible (which is true for relatively dilute electrolyte solutions [61]), then for a given zeolite composition, the ratios of activity coefficients fi ,gi will hardly changein value as the total concentration of electrolyte in the external solution ischanged [56, 60, 62]. Providing these assumptions hold, then taking as an example a binary exchange process, from Eqs. (8b) and (9), it follows that [62]:1

–where the subscripted EA in parentheses indicates that the values of the corrected selectivity coefficient and the rational activity coefficients refer to a particu– –lar composition EB , EA and must be invariant since all the terms of the righthand side are constant or hardly change when the total concentration of theexternal electrolyte solution is changed. The details of the methods which mustbe employed to predict selectivity trends are described elsewhere [62]; theimportant point to note is that if the above assumptions hold then for successfulpredictions it is only required to evaluate the appropriate corrected selectivitycoefficient as a function of zeolite phase composition and to have an accurateknowledge of the solution phase activity coefficient g [54, 55, 62]. For binaryexchanges, this approach has been used to test a variety of systems over the lastdecade, including exchanges involving Pb/Na, Pb/NH4 , Cd/Na and Cd/NH4equilibria in clinoptilolite, ferrierite and mordenite [63–65] using different coanions (chloride, nitrate and perchlorate [62, 66]) as well as the Cd/Na-X andCd/K-X systems [67], with a high level of predictive success [62, 67]. Recently, arelated model has been used with good accuracy for the prediction of K/Na andCa/Na equilibria over a wide range of total ionic concentrations in solution fornatural clinoptilolite [68]. Successful predictions were also achieved for theCa/Na, Ca/Mg and Mg/Na systems in zeolite A [18, 20, 69]; however, for Mg/Naand Mg/NH4 exchanges in a range of faujasites [70], predictions failed badly insome cases. The failures were attributed at the time to salt imbibition, but further detailed experimental studies involving hydronium exchange in the Ca/NaX, Ca/Na-Y, Cs/Na-MOR and Cs/K-MOR systems [71–75] have shown that thesituation is in reality much less straightforward. Failures in predictive methods,particularly at trace levels of exchange, cannot be attributed simply to hydrolysis, hydronium exchange or salt imbibition despite earlier suggestions to thiseffect [70, 76]. An important factor appears to be the presence of colloid-sizezeolite particles [74]. These matters are discussed further in Sect. 3.2.3.To apply the same prediction procedure as that described above for ternaryor multicomponent exchanges, it is helpful to derive analogous equations tothose shown in Eqs. (8), (9) and (10) for binary exchange. For ternary exchange,this was done by Fletcher and Townsend [77] and this approach was used topredict compositions for Na/Ca/Mg-A [20, 69], Na/K/Cd-X [67] and Na/NH4/Mg-X,Y ternary equilibria [70]. For the first two of these systems, ternaryexchange equilibria were predicted successfully but for the Na/NH4/Mg-X,Y systems, the procedure failed for the higher silica Y materials, as for the corresponding conjugate binary exchanges [70]. In parallel with these studies, the

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R.P. Townsend · R. Harjula

model of ternary ion exchange in zeolites [77] was compared with other modelspublished in the literature for clay minerals and resins [78, 79] and a furtherdetailed study [80] came to the conclusion that these other approaches wereappropriate under certain specified conditions [80] for the prediction of exchange equilibria in zeolites.A recent criticism of the ternary exchange model [81], on the basis that theequations could have simply been built up from the conjugate binary systems(obviously true), overlooks the main point. If one uses the conjugate binary systems it is necessary to use a model-based approach to predict activity coefficients for the multicomponent exchange equilibrium in the zeolite and the presence of sublattices within the zeolite framework can make this more difficult todo than for clay minerals and resins (Sect. 1.2) [80]. The ternary exchange model of Fletcher and Townsend [77] does not require one to measure at all the activity coefficients, let alone predict them for multicomponent systems from binarydata, using some model. All that is required is knowledge of the ternary corrected selectivity coefficients that are obtained by integrating the appropriateGibbs-Duhem equations over the ternary composition surface [77] in analogywith the binary approach pioneered by Gaines and Thomas [58]. However,acquiring sufficient data for a ternary system is a difficult and time-consumingexercise [20, 67, 70, 82] and simpler approaches can prove quite adequate provided one validates some of the predictions made [83]. Thus, another model,developed originally for clay minerals [84], has been shown after minor revisionto work well for ternary anion [85] and cation [86] exchanges in organic resinsand has even been extended successfully to a five-component zeolitic system(Sr/Cs/Ca/Mg/Na equilibria in chabazite) [87]. This system is very important inthe field of nuclear waste treatment [87].Accurate prediction is similarly much needed for detergent applications [2, 7,18, 69]. The level and nature of “hardness” in household water varies extensively from one location to another, as do the conditions under which consumersexpect effective laundering to occur (e.g. temperature). Thus accurate selectivitydata (i.e. isotherms and selectivity plots as a function of loading), and reliablepredictive models that are simple to use, are important, since it would clearly beimpossible to measure directly the performance of a given “builder” zeolite forall conceivable situations. Successful predictions have been achieved for thebinary Na/Ca-A, Na/Mg-A and Ca/Mg-A systems [2, 18, 69] as well as for the corresponding ternary system [2, 69]. Similar successful predictions were recentlyachieved also for zeolite MAP [7] once the original iterative procedures ofFranklin and Townsend [69] had been modified appropriately. Figures 7 and 8show examples of such successful predictions in A, for both the binary andternary cases.Occasionally, isotherms of binary and multicomponent exchanges aredescribed using various empirical adsorption equations. These cannot be usedfor the prediction of multicomponent equilibria [88]. In fact, a closer inspectionof these equations reveals that they have no in-built facility for true prediction(i.e. for the calculation of equilibria over ranges of different total solutionconcentrations for heterovalent exchanges). Thus these equations are usefulin describing the observed isotherm in a mathematical form but the only pre-

diction these equations can give is the interpolation of the isotherm underone given set of experimental conditions. With such limited utility, these empirical approaches are not recommended for the “prediction” of ion exchange equilibria.2.4Kinetic Processes and the Prediction of Rates of Exchange

In direct applications involving zeolites as ion exchangers, it is not normally thecase that the system is allowed to reach equilibrium. In batch operations (e.g. indetergency) the time available may be such that the exchange process is interrupted long before equilibrium is reached. Similarly, in column operations (e.g.effluent purification), when the system is operating under steady-state conditions, the balance between throughput of liquid and time of exchange means

Ion Exchange in Molecular Sieves by Conventional Techniques

21

that the system is frequently operating under non-equilibrium conditions.Knowledge of the kinetics of the multicomponent exchange processes (i.e. all ofthe reaction rates, diffusive mechanisms and hydrodynamic processes whichcontribute to the overall rates of exchange of all of the different types of ionsinvolved) is therefore of key importance if one is to be able to predict and control behaviour. Unfortunately, this is easier said than done. The kinetics of ionexchange processes in zeolites are extremely complicated even when one focuses on just one mechanistic process [45]; only recently, it was rightly stated thatthe “picture presented in the literature for diffusion in zeolites is confusing, conflicting and/or inconsistent with theory” [89]. Space permits only a briefoverview of the current state of affairs and this is presented here using a hierarchical model [90] for the zeolite particle or pellet. Much more detail is given elsewhere [45].2.4.1Hierarchical Model of Zeolite Particle or Pellet

Whether one is considering an agglomerate of aggregated zeolite crystallites, ora pellet, a hierarchical model [89, 90] allows one to distinguish the differenttransport and/or rate processes which operate at different length scales.The highest level is concerned with the macroparticle or pellet itself; and thekey issue here is whether transport of ions through the fluid film which encompasses the macroparticle is rate-controlling or not. That this process can be ratecontrolling has been recognised for a long time, being favoured by a low concentration of exchanging ions in solution and a small mean particle size; however, it is known that the hydrodynamic regime pertaining can affect its influence markedly, with high levels of agitation (such as are achieved at highimpeller speeds in a batch reactor [89]) rendering relatively insignificant anymass transfer resistance through the boundary film. The mechanical integrity ofthe macroparticle can also be very important. Taking detergent powder particlesas an example [which can comprise agglomerates of (primary) zeolite crystalline particles held together by means of adhesive, viscoelastic surfactantbridges], these are designed to break up under shear and/or other hydrodynamic regimes that are imposed as part of the wash cycle. On breaking up and dispersing, some of these dispersed smaller particles may find themselves inregions of low agitation and consequently the rate of removal of hardness ionsfrom the wash liquor can be slower than desired due to the onset of film diffusion control.Generally, however, the aim is to avoid conditions leading to film diffusioncontrol. This means that the focus is shifted towards transport processes thatoccur at the intermediate level (that is, in the mesopores and macropores within the macroparticle or pellet itself) and those which occur at the smallestdimensional level (viz., in the very micropores of the molecular sieve) [45,89]. Within the mesopores and macropores between the primary zeolite crystallites transport will be dominated by molecular and ionic intercrystalline diffusion possibly coupled to surface diffusion processes, while, in the zeolitemicropores themselves, intracrystalline diffusion occurs, also possibly coupled

22

R.P. Townsend · R. Harjula

with specific exchange rates associated with the different zeolite sublattices[91, 92].The overall observed kinetics of exchange is of course the result of all theabove-described mechanisms working in concert [45, 89]. To cope with the complexities of the system, a simple approach one may adopt is the homogeneousdiffusion model, which assumes that the behaviour of each distinct diffusingspecies within the macroparticle may be described in terms of a single solidphase “effective diffusivity” [89]. More sophisticated approaches include theheterogeneous diffusion models, where the macropore and micropore diffusionprocesses are described separately and are then assumed in different mathematical treatments either to occur in series or in parallel [45, 89].In practice, to date, most research activity has focused on the intraparticulardiffusion which takes place in the zeolite micropores themselves, on the questionable assumption that these processes are normally the rate-controlling ones.2.4.2Intraparticular Exchange Rate Processes

Our understanding of the processes which govern the rates of ion exchangewithin the micropores of molecular sieves has advanced little over the lastdecade, yet the imperative to be able to control and manipulate these ratesremains as strong as ever. To summarise the current situation it is necessary firstto emphasise some basic principles and then to define certain terms and coefficients.To begin, it is important to distinguish the intrinsic dynamic nature of thesystem from the kinetic processes we actually observe during an ion exchangereaction. An obvious yet important point to remember is that even afterexchange equilibrium has been attained, the equilibrium is a dynamic one. Thustransport of all exchangeable cations and of the solvent molecules continues butafter equilibrium has been reached there are no net changes in the relative distribution of species between, and hence concentrations in, phases with time.This dynamic character is readily verified by adding to the equilibrated systema trace amount of a radioactive isotope of one of the cation types into (say) thesolution phase of the system and then observing the rate at which isotopicexchange between the two phases takes place. The isotopic exchange processmay include as a rate-determining step an intracrystalline exchange process [91,92] but it is also certainly a transport process, which is described in terms of aself-diffusion coefficient D*AA [93]. Self-diffusion coefficients D*AA and D*BB ,which can change markedly with temperature [45] or as the equilibrium concentrations of different cations within the zeolite are altered [45, 94], should besharply distinguished from the exchange diffusion coefficient DAB [95]. DABdescribes the kinetics of the A/B exchange process, that is, the observed rates ofchange of concentrations of ions A and B within each phase as a function of timeand as the system moves to equilibrium.Consider therefore a binary A/B exchange between the zeolite and externalsolution, which is not initially at equilibrium. On mixing the two phases, the Aand B cations, which will almost certainly possess different ionic radii and pos-

Ion Exchange in Molecular Sieves by Conventional Techniques

23

sibly charge, will begin to move in their respective directions of negative chemical potential gradient in order to equalise their respective chemical potentialswithin all phases in the system. However, the mobilities of the two cation typesA and B are likely to be different, which means that the more mobile cation typewill tend to build its concentration, and hence lower its concentration gradient,faster than the other. If this process were to continue unchecked, charge separation within each phase and between the phases would occur, with a concomitantelectrical potential gradient. In practice, of course, the electrical potential gradient that forms as charge separation takes place does not build, but rather acts toslow the faster moving cations and speed the slower ones. Thus it is not adequateto consider only the chemical potential gradients. The net flux JA of (say) the Aexchanging species is actually described by:(11)J = – D [grad c– – (z c– F/RT) grad V]A

AB

A

A A

where F is the Faraday constant and V the electrical potential. An expression forDAB has been derived by Barrer and Rees using an irreversible thermodynamicapproach. The form of this is complicated but, if cross-coefficients otherthan those due to the electrical potential gradient are assumed to be negligible,then [96]:– 2––– 2––D*AA D*BB [c Az A(∂ ln a B / ∂ ln c B) + c B z B (∂ ln a A/ ∂ ln c A)]DAB = 00000000(12)0.– 2– 2D*AAc AzA + D*BBc BzBTwo points should be noted from Eq. (12). First, the magnitude of DAB dependsstrongly on the composition of the exchanger not only because it is a direct function of ionic concentrations, but also because it is a function of both D*AA andD*BB, which we have already noted vary with exchanger composition [45]. Secondly, DAB is a function of the non-ideality of the zeolite [data for which can beobtained, as we saw earlier, from the activity coefficients described in Eq. (10)].One may expect therefore that to describe adequately the kinetic behaviour ofeven a binary exchange process in a molecular sieve would be a very complicated task.To validate this and other similar models, it is necessary to solve, using appropriate boundary conditions, the differential equations describing overall thetransient diffusion process for each ion, of the general form:(13)(∂ c– /∂ t) = div D gradc–i

As an example of the above approach, Brooke and Rees [95] studied the Sr/Cachabazite system. Figure 9 shows their computed time-dependent concentrationprofiles within the zeolite particles both before and after non-ideal behaviourwas taken into account. The effect on DAB of taking non-ideality into account waseven more dramatic, with a discontinuity appearing in the plot of ∂DAB / ∂ c–A